It's been a week since Caroline Kennedy's name vaulted to the top of the list of candidates to replace Hillary Rodham Clinton in the Senate, and already the bloom is coming off the boomlet.

Kennedy, the 51-year-old lawyer, education advocate and daughter of President John F. Kennedy, has done little more than express interest in the job during a phone call with Gov. David Paterson — but that doesn't mean the woman who inspired "Sweet Caroline" is being spared the elbow-to-the-teeth New York treatment.

Rivals — including at least three members of the New York congressional delegation — are starting to doubt Kennedy's viability and experience, and Paterson is said to be less than enthusiastic about picking her, people close to him say. Critics are even questioning the substance of her accomplishments in education, her most high profile issue.

"There is no frontrunner - period," snapped a person close to Paterson, when asked about Kennedy's prospects.

Kennedy faces several obstacles for her candidacy. She has a low public profile in New York, and beyond her name, the most robust element of her public record is a part-time, unpaid fundraising position in New York Mayor Mike Bloomberg's Department of Education.

She's never demonstrated much appetite for the folksy, repetitive rituals of retail politics. The most potent argument being used against her, however, is the little matter of Kennedy's backing of Barack Obama in the primary over New York's native (if adopted) daughter Clinton, a genteel betrayal many Clinton supporters haven't quite forgotten.

"This isn't a jihad or anything, but I'd be lying to you if I said that supporters of Hillary don't remember where she was in the primary," said Democratic Rep. Anthony Weiner, a possible mayoral candidate who is close to the former first lady.

A senior Democrat said Clinton supporters had expressed their resentment over a possible Kennedy pick to people around Paterson, and that tacit resistance has opened the door to other members of the congressional delegation to violate party taboos and directly criticize a Kennedy.

"I don't know what Caroline Kennedy's qualifications are, except that she has name recognition, but so does J.Lo," another Queens congressman, Gary Ackerman, quipped during a radio interview Monday. "I wouldn't make J.Lo the senator unless she proved she had great qualifications, but we haven't seen them yet."

Manhattan Rep. Carolyn Maloney, who has hired a one-time Paterson consultant to lobby for the Senate appointment, was overheard complaining about Kennedy's "complete lack of experience" to friends on the floor of the House on Wednesday, according to a member who was within earshot.

Kennedy worked briefly as a journalist. She has a law degree, but has not practiced extensively. She has co-written two books and edited others, raised money for charities, and served at times as a family spokeswoman, with her highest profile role coming earlier this year when she bestowed the Kennedy mantle on Obama.

"The question is experience," said Hank Sheinkopf, a veteran New York Democratic consultant. "Can she do what needs to be done?"

He also questioned whether the Kennedy name, so potent in the past, would enthrall 21st century New Yorkers.

"So they decided to rename the [Triboro] Bridge [after Robert F. Kennedy]," he said. "But if you went to the street and asked ten people under 50 who this guy was, they wouldn't know what you're talking about."

All of this is new to Kennedy, who has spent an uncommonly dignified adulthood in Manhattan, rearing her three children, quietly taking the Lexington subway line to and from work, raising millions for charitable causes and, most recently, devoting her energies to a public-private partnership that has raised $65 million for New York City public schools.

The attacks on Kennedy were probably inevitable, considering that her emergence as a contender immediately overshadowed a dozen or so toiling lesser-knowns who saw Clinton's seat as the reward for careers spent on the Empire State's political greased pole, the only path to power available to non-Kennedys.

Kennedy's reputation for personal, low-key decency make her a difficult target. Her greatest admirer is Mayor Michael Bloomberg, who told reporters last week that, "Caroline Kennedy is a very experienced woman. She's worked very hard for the city. I can just tell you she's made an enormous difference in New York City."

Another booster is Washington Post columnist Ruth Marcus, who recently enthused: "On the question of Caroline Kennedy for Senate, my head says no, on balance. My heart says yes! Yes! ... What really draws me to the notion of Caroline as senator, though, is the modern fairy tale quality of it all."

And former sports columnist Mike Lupica of the Daily News, the tabloid that has long appealed to the city's cops and firefighters, praised her "quiet grace."

Still in doubt, however, is the substance of her record on education, which seems to be the central piece of her recent résumé. From 2002 to 2004, Kennedy worked as chief executive for the Office of Strategic Partnerships for the New York City Department of Education. During this time, she helped raise more than $65 million for the city's public schools.

She currently serves as vice chairwoman of The Fund for Public Schools, a public-private partnership founded in 2002 to attract private funding for public schools in New York City. A former colleague at the Department of Education described Kennedy as a pleasant, low-key part-timer who appeared at the agency's downtown office for partial days two or three times a week and sat in on meetings, but was rarely a driving force, and who insistently avoided media attention and interview requests.

Her appointment to this job raised eyebrows at the time because it was structured to avoid her having to disclose her family income on city conflict of interest forms that are required for all city officials.

"She wasn't in a paid position, and she wasn't even in any official unpaid position," said Wayne Hawley, general counsel for the Conflict of Interest Board, "She was the board chair of a not-for-profit affiliated with the Board of Education."

Eva Moskowitz, former chairwoman of the City Council's education committee, said Kennedy didn't play a major role in forming city schools policy at the department. Still, Moskowitz, who now runs a school in Harlem, would occasionally run into Kennedy — usually on the subway — and the pair would quickly become engrossed in discussions about education reform and anti-poverty policy.

"I wouldn't say she was a wonk at all," says Moskowitz, "But she was genuinely thoughtful and interested. I would classify her as a progressive."

A spokeswoman for Kennedy didn't respond a request for an interview.

Ultimately, though, the choice will be Paterson's, a free agent in New York politics whose move nobody pretends to predict. One hint was Paterson's praise, as the Kennedy chatter peaked, of incoming American Federation of Teachers President Randi Weingarten, whose name had just surfaced in connection with the seat. Weingarten is Kennedy's opposite: A tough, street-smart lesbian labor lawyer deeply rooted in urban politics, and the flotation of her name appeared aimed at defusing the Kennedy buzz.

Her name also appears to have come from Paterson himself: A person familiar with Weingarten's conversation with Paterson said the governor raised the possibility of the Senate seat.

What the swelling resistance to Kennedy's candidacy makes clear is that she will not inherit the seat. If she wants it, she'll have to fight for it. And to fight for it, she needs to want it — and even her supporters aren’t sure she does.

"She would make a fine senator," said former Mayor Ed Koch. "I don't think it'll happen because she has also made clear that her family takes her entire attention. She's the kind of person who comes in and out of politics on her schedule and for brief periods of time."

Said Weiner, the Clinton backer and Kennedy skeptic: "I do think you have to not only be willing to be milking cows at the state fair, but you've got to like it or at least be very good at acting like you like it," he said. "If she has the gift of milking cows, it's been utterly hidden from people of the state of New York."